INTEGERS 14 (2014)
#A26
ADDITIVE AND MULTIPLICATIVE STRUCTURES OF C ? -SETS
Dibyendu De1
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India
dibyendude@klyuniv.ac.in
Received: 6/14/13, Revised: 3/17/14, Accepted: 4/6/14, Published: 5/30/14
Abstract
It is known that for an IP? set A in (N, +) and a sequence hxn i1
n=1 in N, there exists
1
1
1
a sum subsystem hyn i1
of
hx
i
such
that
F
S(hy
i
)
n n=1
n n=1 [ F P (hyn in=1 ) ✓ A.
n=1
?
Similar types of results have also been proved for central sets where the sequences
have been considered from the class of minimal sequences. In this present work, we
shall prove some analogous results for C? -sets for a more general class of sequences.
1. Introduction
A famous Ramsey-theoretic result is Hindman’s Theorem:
Sr
Theorem 1.1. Given a finite coloring of N = i=1 Ai , there exists a sequence
hxn i1
n=1 in N and i 2 {1, 2, . . . , r} such that
(
)
X
1
F S(hxn in=1 ) =
xn : F 2 Pf (N) ✓ Ai ,
n2F
where for any set X, Pf (X) is the set of all finite nonempty subsets of X.
A strongly negative answer to a combined additive and multiplicative version
of Hindman’s Theorem was presented in [12, Theorem 2.11]. Given a sequence
hxn i1
n=1 in N, let
P S(hxn i1
n=1 ) = {xm + xn : m, n 2 N, m 6= n}
and
P P (hxn i1
n=1 ) = {xm · xn : m, n 2 N, m 6= n}.
Theorem 1.2. There exists a finite partition R of N with no one-to-one sequence
1
1
hxn i1
n=1 in N such that P S(hxn in=1 ) [ P P (hxn in=1 ) is contained in one cell of the
partition R.
1 The
author acknowledges the support received from the DST-PURSE programme grant.
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The original proof of Theorem 1.1 was combinatorial in nature. But later, using
the algebraic structure of N, a very elegant proof of Hindman’s Theorem was
established by Galvin and Glazer, which they never published. A proof of the
Theorem 1.1, that uses the algebraic structure of N was first presented in [6,
Theorem 10.3]. One can also see the proof in [14, Corollary 5.10].
Let us first give a brief description of the algebraic structure of Sd for a discrete
semigroup (S, ·). We take the points of Sd to be the ultrafilters on S, identifying
the principal ultrafilters with the points of S and thus pretending that S ✓ Sd .
Given a set A ✓ S, let us define the subsets of Sd by the following formula:
c`A = A = {p 2 Sd : A 2 p}.
Then the set {c`A ✓ Sd : A ✓ N} forms a basis for the closed sets of Sd as well
as for the open sets. The operation “ · ” on S can be extended to the Stone-Čech
compactification Sd of S, so that ( Sd , ·) becomes a compact right topological
semigroup (meaning that for any p 2 Sd , the function ⇢p : Sd ! Sd , defined by
⇢p (q) = q · p, is continuous) with S contained in its topological center (meaning that
for any x 2 S, the function x : Sd ! Sd , defined by x (q) = x · q, is continuous).
A nonempty subset I of a semigroup T is called a left ideal of S if T I ⇢ I, a right
ideal if IT ⇢ I, and a two-sided ideal (or simply an ideal) if it is both a left and a
right ideal. A minimal left ideal is a left ideal that does not contain any proper left
ideal. Similarly, we can define a minimal right ideal and the smallest ideal.
Numakura proved in [18] (as remarked in [10, Lemma 8.4]), that any compact
Hausdor↵ right topological semigroup T contains idempotents and therefore has a
smallest two-sided ideal
K(T ) =
=
S
S
{L : L is a minimal left ideal of T }
{R : R is a minimal right ideal of T }.
Given a minimal left ideal L and a minimal right ideal R, it easily follows that
L \ R is a group and thus in particular contains an idempotent. If p and q are
idempotents in T , we write p  q if and only if p · q = q · p = p. An idempotent is
minimal with respect to this relation if and only if it is a member of the smallest
ideal K(T ) of T .
Given p, q 2 S and A ✓ S, the set A 2 p·q if and only if {x 2 S : x 1 A 2 q} 2 p,
where x 1 A = {y 2 S : x · y 2 A}. See [14] for an elementary introduction to the
algebra of S and for any unfamiliar details.
A set A ✓ N is called an IP? set if it belongs to every idempotent in N. Given
1
a sequence hxn i1
n=1 in N, we let F P (hxn in=1 ) be the product analogue of finite
1
sums. Given a sequence hxn in=1 in N, we say that hyn i1
n=1 is a sum subsystem
1
of hxn i1
,
provided
there
is
a
sequence
hH
i
of
nonempty
finite subsets of N,
n=1
Pn n=1
such that max Hn < min Hn+1 and yn = t2Hn xt for each n 2 N. The following
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Theorem [5, Theorem 2.6] shows that IP? sets have substantially rich multiplicative
structures.
?
Theorem 1.3. Let hxn i1
n=1 be a sequence in N and A be an IP set in (N, +).
1
1
Then there exists a sum subsystem hyn in=1 of hxn in=1 such that
1
F S(hyn i1
n=1 ) [ F P (hyn in=1 ) ✓ A.
Let us recall the definition of central set [14, Definition 4.42].
Definition 1.4. Let S be a semigroup and C ✓ S. Then C is said to be central if
and only if there is some idempotent p 2 K( S) such that C 2 p.
The algebraic structure of the smallest ideal of S plays a significant role in
Ramsey Theory. It is known that any central subset of (N, +) is guaranteed to have
a substantial amount of additive combinatorial structures. But Theorem 16.27 of
[14] shows that central sets in (N, +) need not admit any multiplicative structure at
all. On the other hand, in [5, Theorem 2.4] we see that sets which belong to every
minimal idempotent of N, called central? sets, must have significant multiplicative
structures. In fact, central? sets in any semigroup (S, ·) are defined to be those sets
which meet every central set.
Theorem 1.5. If A is a central? set in (N, +), then it is also central in (N, ·).
In case of central? sets, a result similar to 1.3 was proved in [7, Theorem 2.4]
for a restricted class of sequences called minimal sequences. Recall that a sequence
hxn i1
n=1 in N is said to be a minimal sequence if
1
\
m=1
c`F S(hxn i1
n=m ) \ K( N) 6= ;.
?
Theorem 1.6. Let hyn i1
n=1 be a minimal sequence and let A be a central set
1
1
in (N, +). Then there exists a sum subsystem hxn in=1 of hyn in=1 such that
1
F S(hxn i1
n=1 ) [ F P (hxn in=1 ) ✓ A.
A similar result was proved in a di↵erent setup in [9].
The original Central Sets Theorem was proved by Furstenberg in [10, Theorem
8.1] (using a di↵erent but equivalent definition of central sets). However, the most
general version of Central Sets Theorem is in [8]. We state it here only for the case
of a commutative semigroup.
Theorem 1.7. Let (S, ·) be a commutative semigroup, and let T = NS be the set
of all sequences in S. Let C be a central subset of S. Then there exist functions
↵ : Pf (T ) ! S and H : Pf (T ) ! Pf (N) such that
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1. if F, G 2 Pf (T ) and F ( G then max H(F ) < min H(G), and
2. whenever m 2 N, G1 , G2 , . . . , Gm 2 Pf (T ), G1 ( G2 ( . . . ( Gm , and for
each i 2 {1, 2, . . . , m}, fi 2 Gi , one has
m
Y
i=1
↵(Gi ) ·
Y
t2H(Gi )
fi (t) 2 C.
Recently, a lot of attention has been paid to those sets which satisfy the conclusion of the latest Central Sets Theorem.
Definition 1.8. Let (S, ·) be a commutative semigroup, and let T = NS be the set
of all sequences in S. A subset C of S is said to be a C-set if there exist functions
↵ : Pf (T ) ! S and H : Pf (T ) ! Pf (N) such that
1. if F, G 2 Pf (T ) and F ( G, then max H(F ) < min H(G) and
2. whenever m 2 N, G1 , G2 , . . . , Gm 2 Pf (T ), G1 ( G2 ( . . . ( Gm , and for
each i 2 {1, 2, . . . , m}, fi 2 Gi , one has
m
Y
i=1
↵(Gi ) ·
Y
t2H(Gi )
fi (t) 2 C.
We now present some notation from [8].
Definition 1.9. Let (S, ·) be a commutative semigroup, and let T = NS be the set
of all sequences in S.
1. A subset A of S is said to be a J-set if for every F 2 Pf (T ) there exist a 2 S
and H 2 Pf (N) such that for all f 2 F ,
Q
a · t2H f (t) 2 A.
2. J(S) = {p 2 S : (8A 2 p)(A is a J-set)}.
Theorem 1.10. Let (S, ·) be a discrete commutative semigroup and A be a subset
of S. Then A is a J-set if and only if J(S) \ c`A 6= ;.
Proof. Since the collection of J-sets forms a partition regular family, the theorem
follows from [14, Therem 3.11].
The following is a consequence of [8, Theorem 3.8]. The easy proof for the
commutative case can be found in [15, Theorem 2.5].
Theorem 1.11. Let (S, +) be a commutative semigroup and let T = NS be the set
of all sequences in S, and let A ✓ S. Then A is a C-set if and only if there is an
idempotent p 2 c`A \ J(S).
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We conclude these introductory discussions with the following [8, Theorem 3.5].
Theorem 1.12. If (S, ·) be a discrete commutative semigroup then J(S) is a closed
two-sided ideal of S and c`K( S) ⇢ J(S).
2. C ? -set
We have already discussed IP? and central? sets before; let us now introduce the
notion of C ? -set.
Definition 2.1. Let (S, ·) be a discrete commutative semigroup. A set A ✓ S is
said to be a C? -set if it is a member of all the idempotents of J(S).
It is clear from the definition of C? -set that
IP? -set ) C? -set ) central? -set.
Remark 2.2. It is shown in [13] that there exists a set A ⇢ N which is a C-set in
(N, +), but its upper Banach density (defined below) vanishes. Since A is a C-set
in (N, +), there exists an idempotent p 2 J(N) such that A 2 p. But, as the upper
Banach density of A is zero, it is not a central set in (N, +), so it is not contained
in any minimal idempotent of N. Hence N \ A is a member of all the minimal
idempotents in N. Therefore N \ A is a central? -set but not a C? -set as N \ A 62 p.
Definition 2.3. Let (S, +) be a discrete commutative semigroup. A sequence
hxn i1
n=1 is said to be an almost minimal sequence if
1
\
m=1
c`F S(hxn i1
n=m ) \ J(S)) 6= ;.
To provide an example of an almost minimal sequence which is not minimal, let
us recall the following notion of “Banach density”. The author is grateful to Prof.
Neil Hindman for his help in constructing this example.
Definition 2.4. Let A ⇢ N. Then
1. d⇤ (A) = sup{↵ 2 R : (8 k 2 N)(9n > k)(9a 2 N)(|A\{a+1, a+2, . . . , a+n}|
↵ · n)}.
2. 4⇤ = {p 2 N : (8A 2 p)(d⇤ (A) > 0)}.
d⇤ (A) is said to be the upper Banach density of A.
It follows from [14, Theorem 20.5 and 20.6] that 4⇤ is a closed two-sided ideal
of ( N, +), so that c`K( N) ⇢ 4⇤ .
Let us now recall the following Theorem from [1], which we shall require to
construct example of an almost minimal sequence which is not minimal.
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Theorem 2.5. Let hxn i1
n=1Tbe a sequence in N, such that for all n 2 N we have
Pn
1
xn+1 > t=1 xt , and T = m=1 c`F S(hxn i1
n=m ). Then the following conditions
are equivalent:
1. T \ K( N) 6= ;;
2. T \ c`K( N) 6= ;;
Pn
3. the set {xn+1
t=1 xt : n 2 N} is bounded.
Proof. See [1, Theorem 2.1].
We already know that a subset A of N is central if it belongs to some idempotent
of K( N). Further, the members of the ultrafilters of K( N) are piecewise syndetic.
Replacing piecewise syndeticity with positive upper Banach density leads to the
class of essential idempotents: viz. q 2 N is an essential idempotent if it is an
idempotent ultrafilter, all of whose elements have positive upper Banach densitythat is, q 2 4⇤ . By [4, Theorem 2.8], a set S ✓ N is a D-set if it is contained in
some essential idempotent. The authors proved in [3, Theorem 11] that D-sets also
satisfy the conclusion of the original Central Sets Theorem and are in particular
J-sets.
Now let d 2 N, and let us define a sequence hxn i1
follown=1 in N by the
Pn
Pn
n+(d 1)
ing formula: xn+1 =
c. Then the set {xn+1
t=1 xt + b T d
t=1 xt :
1
n 2 N} being unbounded, we have m=1 c`F S(hxn i1
n=m ) \ K( N) = ;. Therefore, the sequence hxn i1
But by [1, Lemma 2.20], we have
n=1 is not minimal.
T1
T1
1
⇤
c`F
S(hx
i
)\4
=
6
;.
Since
c`F
S(hxn i1
)\4⇤ is a compact subn n=m
m=1
m=1
T1n=m
⇤
semigroup of N, we can choose an idempotent p in m=1 c`F S(hxn i1
n=m ) \ 4 . In
1
1
particular, F S(hxn in=1 ) 2 p. Therefore, by the above discussion, F S(hxn in=1 ) satisfies the conclusion of the original Central Sets Theorem and is in particular a J-set.
T1
Hence by the foregoing Theorem 2.7, we have m=1 c`F S(hxn i1
n=m ) \ J(N)) 6= ;.
This implies that this sequence hxn i1
is
almost
minimal.
n=1
Question 2.6. Does there exist a sequence hxn i1
n=1 in N with the property that
F S(hxn i1
n=1 ) is a C-set but its Banach density is zero?
The author is thankful to the anonymous referee for her/his help in simplifying
the following proof.
Theorem 2.7. In the semigroup (N, +), the following conditions are equivalent:
(a) hxn i1
n=1 is an almost minimal sequence;
(b) F S(hxn i1
n=1 ) is a J-set;
(c) there is an idempotent in
T1
m=1
c`F S(hxn i1
n=m ) \ J(N)) 6= ;.
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Proof. (a) ) (b): The proof follows from the definition of an almost minimal sequences.
(b) ) (c): Let F S(hxn i1
n=1 ) be a J-set. Then by Theorem 1.10, we have J(N) \
c`F S(hxn i1
)
=
6
;.
We
choose
p 2 J(N) \ c`F S(hxn i1
n=1
n=1 ). By [14, Lemma 5.11],
T1
1
m=1 c`F S(hxn in=m ) is a subsemigroup of N. As a consequence of [8, Theorem
3.5], it follows that J(N) is a subsemigroup of N. Also, J(N) being closed, it
is a compact subsemigroup of N. Therefore, it suffices to show that for each m,
T1
1
m=1 c`F S(hxn in=m )\J(N) 6= ;, because then it must contain an idempotent. For
this, it in turn suffices to show that, for any m 2 N we have c`F S(hxn i1
n=m )\J(N) 6=
1
1
1
;. So let m 2 N be given. Then F S(hxn in=1 ) = F S(hxn in=m ) [ F S(hxn im
n=1 ) [
S
m 1
1
{t + F S(hxn in=m ) : t 2 F S(hxn in=1 )}. Hence we must have one of the following:
1
1. F S(hxn im
n=1 ) 2 p;
2. F S(hxn i1
n=m ) 2 p;
m 1
3. t + F S(hxn i1
n=m ) 2 p for some t 2 F S(hxn in=1 ).
Clearly (1) does not hold, because in that case p becomes a member of N, that is
a principal ultrafilter, while p 2 N \ N. If (2) holds, then we are done. So assume
1
1
that (3) holds. Then for some t 2 F S(hxn im
n=1 ), we have that t+F S(hxn in=m ) 2 p.
We choose some q 2 c`F S(hxn i1
n=m ) so that t + q = p. For every F 2 q, we have t 2
{n 2 N : n + (t + F ) 2 q} so that t + F 2 p. Since J-sets in (N , +) are translation
invariant, F becomes a J-set. Thus q 2 J(N) \ c`F S(hxn i1
n=m ).
(c) ) (a): This case is obvious.
Lemma 2.8. If A is a C-set in (N, +) then for any n 2 N, nA and n
C-sets, where n 1 A = {m 2 N : n · m 2 A}.
1
A are also
Proof. [17, Lemma 8.1].
Lemma 2.9. If A is a C ? -set in (N, +) then n
1
A is also a C ? -set for any n 2 N.
Proof. Let A be a C ? -set and t 2 N. To prove that t 1 A is a C ? -set, it is sufficient
to show that for any C-set C, we have C \ t 1 A 6= ;. Since C is a C-set, tC is
also a C-set, so that A \ tC 6= ;. Choose n 2 tC \ A and k 2 C such that n = tk.
Therefore k = n/t 2 t 1 A. Hence C \ t 1 A 6= ;.
?
Theorem 2.10. Let hxn i1
n=1 be an almost minimal sequence and A be a C -set in
1
(N, +). Then there exists a sum subsystem hyn i1
n=1 of hxn in=1 such that
1
F S(hyn i1
n=1 ) [ F P (hyn in=1 ) ✓ A.
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Proof. Since hxn i1
n=1 is an almost minimal sequence, by Theorem 2.7, we can find
some idempotent p 2 J(N) such that F S(hxn i1
n=m ) 2 p, for each m 2 N. Again,
since A is a C ? -set in (N, +), from Lemma 2.9, it follows that, s 1 A 2 p, for every
s 2 N. Let A? = {s 2 A : s + A 2 p}. Then by [14, Lemma 4.14], we have
A? 2 p. Then we can choose y1 2 A? \ F S(hxn i1
n=1 ). Inductively, let m 2 N and
m
hyi im
i=1 , hHi ii=1 in Pf (N) be chosen with the following properties:
1. for all i 2 {1, 2, . . . , m 1}, max Hi < min Hi+1 ;
P
P
?
2. if yi = t2Hi xt then t2Hm xt 2 A? and F P (hyi im
i=1 ) ✓ A .
P
We observe that { t2H xt : H 2 Pf (N), min H > max Hm } 2 p. Let us set
P
B = { t2H xt : H 2 Pf (N), min H > max Hm }, E1 = F S(hyi im
i=1 ) and E2 =
F P (hyi im
i=1 ). Now consider
\
\
D = B \ A? \
( s + A? ) \
(s 1 A? ).
s2E1
s2E2
Then D 2 p. Choose ym+1 2 D and Hm+1 2 Pf (N) such that min Hm+1 >
P
max Hm . Putting ym+1 = t2Hm+1 xt , it shows that the induction can be continued
and this eventually proves the theorem.
Acknowledgements. The author is thankful to Prof. Neil Hindman for his helpful suggestions and comments. The author would also like to thank the anonymous
referee for many helpful suggestions and corrections, and in particular for the assistance she/he provided towards simplifying the proof of Theorem 2.7.
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